Kernel

Definition:
The kernel of a homomorphism f, often denoted by ker f, is the set of elements of G that are mapped to the identity in G': ker f={a is an element of G|f(a)=1}. The kernel is important because it controls the entire homomorphism. It tells us not only which elements of G are mapped to the identity in G', but also which pairs of elements have the same image in G'.